Problem 33 - Algebra 1

Question

Write an equation of the line that is parallel to the given line and passes through the given point. $$y=x+4,(-2,0)$$

Step-by-Step Solution

Answer

The equation of the line that is parallel to \(y=x+4\) and passes through \(-2,0\) is \(y=x+2\).

Step 1: Recall the slope and the point

The slope of the line \(y=x+4\) is 1, and the line passes through the point \(-2,0\).

Step 2: Substitute the values into the equation

The equation of a line in slope-intercept form is \( y = mx + c\), where \(m\) is the slope and \(c\) is the y-intercept. We know \(-2\) and \(0\) are the x and y coordinates of the point. So we substitute them into the equation giving us \(0=1*(-2)+c\).

Step 3: Solve for the y-intercept

Solving the equation for \(c\) gives us \(c = 2\).

Step 4: Write the final equation of the line

So, the equation of the line that is parallel to \(y=x+4\) and passes through \(-2,0\) is \(y=x+2\).

Key Concepts

Slope-Intercept Form

Understanding the slope-intercept form of an equation is crucial when dealing with linear equations. This form is expressed as \( y = mx + c \). Here’s what each part means:

• \( y \): This represents the dependent variable, usually the outcome or output you want to calculate.
• \( m \): The slope of the line, showing how steep the line is and the direction it goes (increasing or decreasing).
• \( x \): The independent variable, which you manipulate to determine \( y \).
• \( c \): The y-intercept, the point where the line crosses the y-axis.

The slope, \( m \), is particularly important because it dictates the angle and direction of the line. For example, a positive slope means the line rises as it moves to the right, while a negative slope means it falls.
Parallel lines share the same slope, which makes their equations straightforward to form once you understand this pattern.

Point-Slope Form

The point-slope form is another useful way to write the equation of a line, particularly when you know a point on the line and the slope. The formula is \( y - y_1 = m(x - x_1) \), which can be broken down into:

• \( y \) and \( x \): Variables representing coordinates on the line.
• \( y_1 \): The y-coordinate of the known point on the line.
• \( x_1 \): The x-coordinate of the known point.
• \( m \): The slope of the line.

This form is particularly handy when you need to find a line parallel to another, as you can directly input the slope and a point through which the line passes. Let’s say you have a line with a slope of 1, like in our equation, \( y = x + 4 \).
If it must pass through the point \((-2, 0)\), using the point-slope form directly can simplify the process of finding the line's equation quickly.

Linear Equations

Linear equations are mathematical statements that describe a straight line when they are graphed. They are the building blocks for understanding more complex algebraic equations and play a crucial role in various problem-solving scenarios. Linear equations follow the general principle of representing a relation between variables with a consistent rate of change, or slope.
Solving these equations often involves isolating the desired variable using algebraic techniques such as addition, subtraction, multiplication, or division. Consider the exercise's last step, where we solved for \( c \) in the equation \( 0 = 1(-2) + c \). This stepwise unraveling is an essential skill in navigating linear equations.
By mastering these basic manipulations, students enhance their ability to manage more complex problems later on, as linear equations form the foundation for the study of algebra and calculus.